Ten riddles that put your math skills to the test
Give your brain a workout with these fun math riddles from Ohio State Professor Jim Fowler, dubbed “the prof who is making calculus go viral” by Forbes.
Forbes magazine called Jim Fowler “the prof who is making calculus go viral.” Now, this assistant professor of mathematics — renowned for his knack for teaching via online courses, video and virtual reality — has a challenge for you: two handfuls of math riddles compiled especially for Ohio State Alumni Magazine readers. Pencils up.
How did I get here?
Pick a positive whole number. If it is odd, multiply by 3 and add 1; if it is even, divide by 2. Repeat. Keep repeating! Did you eventually get back to 1?
Yes, you should get back to 1. That this happens is known as the Collatz conjecture, but it has not been proven — yet.
It’s OK to cut corners
A domino can cover two squares on a chessboard. Assume the one you’re working with has two opposite corners removed. Can you cover the board with dominoes?
No, you can’t completely cover the chessboard. Each domino covers a light square and a dark square, but after removing opposite corners, there are different numbers of light and dark squares.
Give this a try
Can you find a whole number that, when multiplied by itself, ends in a 3?
No whole number, when multiplied by itself, ends in a 3. In fact, the units digits of perfect squares are 0, 1, 4, 5, 6, 9.
OK, world traveler
Pick your five favorite cities. Try to cut the Earth in half so that at least part of four of your cities lie in one half.
Yes. Here’s how: Cut the Earth in half along a great circle through the first two cities; of the other three cities, two are on the same piece.
Pick a number between 1 and 9. Now, raise it to the fifth power by multiplying five copies of your number together. It might be a big number, but what’s the last digit?
It’s the same as the number you started with. Bonus question: Does this work if fifth power is replaced by fourth power?
Party of 6 please
You’re at a dinner party with six guests. Can you always find three people who are mutual friends or mutual strangers?
Yes. In addition to yourself, you’re friends (or strangers) with three of the other five people. Suppose it’s friends. Then if those three people are mutual strangers, you’ve found your three people. If not, one pair of your friends are friends with each other, and with you and that pair you’ve found your three. This is an example of Ramsey theory, a branch of mathematics that studies the conditions under which order must appear.
The chance of regifting
Your friends bring gifts to a large party, and you ungraciously hand the gifts back to them randomly. What’s the chance no guest gets his or her own gift back?
Among a large number of guests, this would happen about 37 percent of the time. Surprisingly, the probability is closer to 1/e as the number of guests gets larger.
A perfect power
Eight is a perfect cube (2 x 2 x 2) and nine is a perfect square (3 x 3). They’re both examples of “perfect powers” (and the only example of consecutive perfect powers among the positive integers, as Preda Mihăilescu proved in 2002). Can you find perfect powers that differ by two?
25 is 5 times 5, and 27 is 3 times 3 times 3.
Nine, you’re a square
Perfect squares are numbers such as 0, 1, 4, 9, 16, 25. Surprisingly, every whole number is the sum of four perfect squares. 71 is 4 + 9 + 9 + 49. Can you write 71 as a sum of four perfect squares?
71 = 1 + 9 + 25 + 36
Awwwww, is that all?
Write down 1 followed by a bunch of zeroes and divide it by 37. What’s the remainder?
The remainder is 1, 10 or 26. Are you surprised that there are only three possibilities?